WASHINGTON — With President Obama insisting on higher tax rates for affluent Americans and winning public support for the idea, congressional Republicans find themselves in an increasingly difficult political spot and are quietly beginning the difficult search for a way out.

Senior Republican leadership aides say they are contemplating a fallback position since a standoff over expiring tax rates will be portrayed by Democrats as evidence that the opposition is willing to allow taxes to rise on the middle class to keep taxes from rising on the rich — and their intransigence could mean taxes go up on rich, poor, and middle class alike.

The leadership officials now say that if no deal can be struck to avert the en masse expiration of all the Bush-era tax cuts and the onset of deep, across-the-board spending cuts, they could foresee taking up and passing legislation this month to extend the expiring middle-class tax cuts, then resume the bitter fight over spending and taxes as the nation approaches the next hard deadline: its statutory borrowing limit, which could be reached in late January or February.

“There’s always better ground, but you have to get there,’’ said Representative Michael C. Burgess, Republican of Texas, who made it clear he does not support allowing any taxes to rise.

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Senator Olympia J. Snowe of Maine, who is retiring, joined a handful of other Republicans on Tuesday suggesting that Congress should pass the middle-class tax cut extensions now, then leave the fight over taxes and spending until later. Americans, she said, ‘‘should not even be questioning that we will ultimately raise taxes on low- to middle-income people.’’ Congress could take that off the table ‘‘while you’re grappling with tax cuts for the wealthy,’’ she said.

But any move toward compromise with Democrats on fiscal issues quickly comes under attack from conservatives as a surrender and unsettles the rank-and-file.

It is a dynamic that has haunted Speaker John A. Boehner throughout the 112th Congress, as he has repeatedly been caught between the imperative to govern and the need to satisfy the restive right. Boehner, of Ohio, has drawn fire this week for removing a handful of House Republicans who have defied leadership from their preferred committee seats, a step he took to enforce party discipline.

Obama made clear on Tuesday in an interview with Bloomberg News that he was not going to budge on raising tax rates on income over $250,000. ‘‘We’re going to have to see the rates on the top 2 percent go up,’’ Obama said in his first interview since his reelection, ‘‘and we’re not going to be able to get a deal without it.’’

The prospect of allowing an extension only of the middle-class tax cuts is just one possibility, and Congress may never reach that point if an agreement can be reached or if another alternative can be found. And it would be a bitter pill for Republicans to swallow since they have repeatedly called for an extension of all the expiring tax cuts, saying any increases could harm the economy.

But Republicans also know they have a problem: many liberal Democrats are more than willing to return to the Clinton-era tax code, and to allow across-the-board cuts to take effect, which disproportionately affect the military, rather than compromise too much with Republicans after the strong Democratic showing in the elections.

“It’s a terrible position because, by default, Democrats get what they want,’’ said Representative James Lankford, Republican of Oklahoma, who admitted his party is boxed in.

But in the debt ceiling fight, the tables might turn. Many conservative Republicans say they are willing to let the nation default if Congress refuses to cut spending. If the tax rate fight is already resolved by the time the debt limit increase is needed, Democrats will find themselves without the leverage they now have with the expiration of the lower tax rates a certainty unless Congress acts affirmatively.

That is why in his opening bid to end the fiscal standoff, Obama proposed a permanent policy change to let the president raise the nation’s borrowing limit on his own — and why Senator Mitch McConnell of Kentucky, the Republican leader, reportedly laughed out loud at the idea.

Boehner took as much fire from conservatives as from Democrats after proposing a deficit-reduction plan that would raise $800 billion in tax revenue over 10 years. Conservative advocacy groups and conservatives on Capitol Hill were united in their condemnation.

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Given the difficulty of compromise, a fallback may emerge as a top option. Republican leaders could take up legislation already passed by the Senate to extend tax cuts on income under $250,000, attach a deferral or cancellation of the automatic spending cuts, and give Obama nothing else, denying requests for increased infrastructure spending, help for homeowners to refinance their mortgages, and extensions of the payroll tax cut and unemployment insurance.

Then Republicans would demand deep concessions on spending and changes to Medicare and Social Security as a price to raise the debt ceiling a few weeks later. Republicans say any such decision to follow that course is still a ways off.

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